No, you don’t have to renounce your US citizenship if you live in Belize. You can remain here as a resident for as long as you care to. At this time you do not even have to renounce your US citizenship if you want to become a Belize Citizen. However, I am not saying that that will always be true because the United States might decide to close that loop-hole. At this time you can have Dual Citizenship for US and Belize. Not true with Canadians as Canada does not allow dual citizenship.
There is no visa in Belize. You just come in to the country as a tourist and have your passport stamped every month for a maximum of six months. You cannot work or donate your time during that time. At the end of six months you have to leave unless you have applied for a residency or a work permit.
(Belize City, Belize, pictured.)